INTERPRETATIONS of EUCLIDEAN GEOMETRY We Know Since Descartes That Points of the Euclidean N-Dimensional Space Can Be Identified

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 322, Number I, November 1990

INTERPRETATIONS OF EUCLIDEAN GEOMETRY

S. SWIERCZKOWSKl

ABSTRACT. Following Tarski, we view n-dimensional Euclidean geometry as a first-order theory En with an infinite set of axioms about the relations of betweenness (among points on a line) and equidistance (among pairs of points), We show that for k < n, En does not admit a k-dimensional interpretation in the theory RCF of real closed fields, and we deduce that En cannot be interpreted r-dimensionally in Es' when r' s < n .

1. INTRODUCTION We know since Descartes that points of the Euclidean n-dimensional space can be identified with n-tuples of real numbers. Moreover, each statement of n-dimensional geometry that involves a variable x representing a point has its counterpart in algebra that is a statement involving n variables XI' '" , xn (the "coordinates" of x). In such a situation, we say (to be made precise later in Definition 2.2) that we have an n-dimensional interpretation of the language of geometry in the language of algebra. In the sequel En will be the first-order theory with equality that has been proposed by Alfred Tarski [T2] for describing the n-dimensional Euclidean geometry. We shall denote by Lp,J the language of En; this is the first-order language whose vocabulary consists of three predicate symbols: = (binary), p (ternary), and J (quaternary). If ~ is the Euclidean n-dimensional space, i.e., the usual model of En whose universe is Rn , and a, b, c, dE Rn , then the meaning of the relations p and J is as follows: p (a, b, c) iff b lies between a and c (possibly b = a or b = c), J (a, b, c, d) iff a is as distant from b as c is from d. As Tarski has shown, his axiomatization of En yields a complete theory. Thus, the theorems of En are just those sentences in Lp,J that are true in ~ . The preceding Cartesian n-dimensional interpretation of the language Lp,J in the language of algebra (see the end of §2, where e stands for equality) is also an interpretation of theories in the sense that the algebraic counterparts of theorems of Ell are theorems of RCF, i.e., of the theory of real closed fields,

Received by the editors November 28, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 03F25; Secondary 51 M05. Key words and phrases. Interpretation, definable set. semialgebraic, Cartesian coordinates.

315

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of which the field of real numbers is one. The language of RCF is the language LOF of ordered fields, i.e., the f.o. language whose vocabulary consists of the symbols: +, -, . (binary functions), <, = (binary predicates), and 0, I (constants). Thus, we have the Cartesian n-dimensional interpretation of Lpo in L OF ' which is an interpretation of the theory En in the theory RCF. Mycielski [M] has conjectured that an interpretation of En in RCF cannot be achieved more economically than in the preceding Cartesian way, that is, using k-tuples of variables for some k < n instead of n-tuples. In 1980, Boffa [B] proved that this is so for the case when n = 2 and k = I. The general conjecture is confirmed by the main result of this paper:

Theorem 1.1. For every n 2 2, there exists a theorem (jJn of En such that for every k < n, the image of (jJn under any k-dimensional interpretation of the language L pJ in the language LOF is not a theorem of ReF. Corollary 1.1. If r . s < n, there does not exist an r-dimensional interpretation of the theory En in the theory Es' Indeed, if such an interpretation were to exist, one could superimpose it with the Cartesian s-dimensional interpretation of Es in RCF, thus obtaining an rs-dimensional interpretation of En in RCF, contrary to Theorem 1.1. Our following proof combines the basic idea of Boffa in [B, p. 123] with a result of Lojasiewicz [L] on triangulation of semialgebraic sets (proved also in [C, VdDl, and H]). In 1984, Szczerba announced in [Sz, p. 677] that he had found the affirmative answer to Mycielski's conjecture; however, his work is not known to us and does not seem to have been published. The author wishes to express his gratitude to Jan Mycielski for fruitful dis- cussions that led to many improvements of this paper.

2. INTERPRETATIONS

Given a formula 0:, we say that a variable Xj is substitutable for a variable Xi in 0: if, by replacing Xi with Xj at any of its free occurrences, we get a free

occurrence of x j ' or, in other words, if no free occurrence of Xi is within the scope of the quantifier ::Ixj . Suppose all the free variables of 0: (i.e., variables that have at least one free occurrence) are among Xi ' .. , ,Xi ' where i < i2 < .,. < in' and let I n l XJ. , •.. ,x . be such that xJ. is substitutable for Xi in 0: for each s = InsJ s 1 , ... , n. Then the result of replacing each X at every free occurrence by Is Xjs will be denoted by:

X X· ] 0: [ II In. X X JI I n If the n-tuple (x , ... , x) coincides with (XI"'" X ) or with (x "'" 'I In n o x n _ l ) , we shall write the above as o:[xjl ' ... , Xj) .

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By a variant of a we shall mean any formula obtained from a by a sequence of replacements as follows:

replace 3xkp by 3xsP [~~] ,

where 3xk P is any subformula of a such that Xs is substitutable for x k in P . Obviously, such replacements do not affect the free occurrences of variables and they do not change the scopes of quantifiers. We assume until the end of this section that L is a relational language (i.e., L has no function symbols) and that xo' x, ' ... is the list of all variables of L. Let L' be any language that contains the variables xo' x, ' .... Definition 2.1. By a one-dimensional interpretation of L in L', we mean a map a 1---+ aO , which assigns to every formula a of L a formula aO of L' so that (i) every variable free in aO is also free in a; (ii) for any n-place predicate symbol p of L and any Xi ' ... ,Xi ' I n (p(x , ... , X)) ° IS. ((p(X, ' ... , Xn)) °), [X , ... , Xl· ], where (p(x, ' ... , xn )0)' il In II n is a variant of (p(x" ... ,xn))o in which Xi, is substitutable for Xs for all s = 1, ... , n;

(iii) if a is of the form i,u, ,u V 1], ,u ---* 1], ,u & 1], 'r/xi,u or 3xi,u, then aO is correspondingly il, ,u0 V 1]0, ,u0 ---* 1]0, pO & 1]0 , 'r/xil or 3xil

Obviously, an interpretation a f-t aO is uniquely determined by its restriction to the atomic formulas p(x , ... , Xl· ) , and thus, up to variants, by its restric- I, n tion to the formulas p(x" ... , xn). The images of the latter can be chosen arbitrarily, provided (i) is satisfied. For example, there is exactly one interpretation of L po in LOF such that:

(P(x, ' x2' x 3)) ° IS (X3 - X2) (X2 - X,) 2: 0, (J(X] , X2' X3' X4))0 is (X2 - X,)2 = (X4 - X3)2, and

(Xl = x 2 )0 is Xl = x 2 • This is also the obvious interpretation of El in ReF. In the sequel, we shall treat equality just as any other binary predicate symbol. In particular, it will not necessarily be assumed that (Xl = x2 )0 is Xl = x 2 • To prevent misunderstandings, we shall use henceforth the letter e to denote the binary predicate symbol for equality in any language L. Thus, for example, to define an L-structure S'I , we shall have to specify also the binary relation on S'I corresponding to e. (This will not, in general, be the relation of equality in the structure S'I .) Remarks. (I) Interpretations can be superimposed, that is, from an interpretation of L in L' and another from L' to L" , we get an interpretation of L in L" .

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(2) For interpreting L in L' it is irrelevant which variables are in these languages, provided all variables of L are also in L' . Given a positive integer k and L as just described, let us denote by L (k) the language such that: (a) the only variables of L (k) are xi}' i, j, = 0, 1, ... ,

(b) there is a one-to-one correspondence p t-+ p(k) between the predicate symbols p in Land p(k) in L(k) such that if p is n-place then p(k) is kn- place. We define a mapping a t-+ a(k) , called the k-spread, from formulas of L to formulas of L (k) , as follows. Given a, one obtains a(k) by replacing each occurrence of QXi in a (where Q is :3 or V) by QXil QXi2 ... QXik ' and every occurrence of Xi' other than in QXi' by Xii ' Xi2 ' ... , x ik . Suppose that L" contains the variables xii; i, j = 0, 1 , 2, .... Then one has interpretations of L (k) in L" .

Definition 2.2. By a k-dimensional interpretation of L in L" , we mean a map a t-+ a * from the formulas of L to the formulas of L" that is the composite of the k-spread a t-+ a(k) with a one-dimensional interpretation fJ t-+ fJo of L(k).ill L" (.l.e., a *.IS ((k))O)a . Obviously, the k-spread is the simplest example of a k-dimensional interpretation. To describe the Cartesian k-dimensional interpretation a t-+ a * of Lpd in L OF ' assume that LOF has all xi); i, j = 0, 1, ... , among its variables. Then a t-+ a * is uniquely determined by the following conditions:

2 (i l ) (fJ(x l , x 2 ' X3))* is the conjunction of the k + k formulas

(X3i - X2)(X2i - Xli);:::: 0; 1::; i::; k, l::;i,j::;k,

k k (i2) (J(xi ' x 2' x 3' X4))* IS L:(X2i - xli = L:(X4i - X3/ ' i=! i=1

3. INTERPRETATIONS WITH PARAMETERS Let L, L' be first-order languages where L is relational and all variables of L are also in L'. We denote by L~ the language obtained by adding to L' new variables PO' PI ' ... , which we shall call parameter or p-variables. Let a t-+ :3pa be the map that assigns to every a in L~ the existential closure of a with respect to all the p-variables. We shall call this map the existential p-closure.

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Definition 3.1. A one-dimensional interpretation with parameters of L in L' is a map a f-+ a(O) from formulas of L to formulas of L' obtained by combining O a one-dimensional interpretation a f-+ a of L in L~ with the existential p- closure and next taking a variant that has no p-variables. (Thus a(O) is a variant without p-variables of 3pao .)

Definition 3.2 [M, M-P-S]. Suppose L has only the variables Xo' Xl ' . .. and L" contains all the variables Xi}; i, j = 0, 1 , .... Then, by a k-dimensional interpretation with parameters of L in L", we mean the superposition of the k-spread of L in L (k) with a one-dimensional interpretation with parameters of L(k) in L".

Our main result (Theorem 1.1) is valid for interpretations with parameters. However, it will suffice to prove this theorem for interpretations without parameters, because of the following known fact. (See a remark in [M, §3, p. 299] concerning theories with selectors [M-V].)

Theorem. Given a k-dimensional interpretation with parameters a f-+ a(*) oj a first-order language L in LOF and a closed Jormula rp oj L such that RC F f-

rp(*) , there exists a k-dimensional interpretation without parameters a f-+ a * oj L in LOF such that RCF f- rp* . In view of this, by an interpretation we shall mean henceforth an interpretation without parameters.

4. STRUCTURES AND DEFINABILITY Given an L-structure Jii' , we shall denote by LIN'I the diagram language of L, i.e., the language obtained by adding to L the elements of the universe 1Jii'1 (or rather, their names) as constants. If a is a formula in L with no free variables other than Xl' ... , xn and ai' ... , an E 1Jii'1 , then the formula a[a l ... an] is closed in LIN'I' and thus it has a logical value in Jii' . We shall indicate the value true by prefixing the formula with Jii' 1= • Definition 4.1. Given an L-structure Jii' , a set S will be called Jii' -definable if S c 1Jii'l n for some n 2: 1 and there are ao' ... , am E 1Jii'1 and a formula a in L with no free variables other than xo' ... , xm' xm+l ' ••. , xm+n such that

(Sl ' ... , sn) E S ¢} Jii' 1= a[ao, ... , am' Sl ' ... , sn] for all Sl' ... , sn E 1Jii'1. We shall call xo' ... , xm the parameter variables, and we shall denote S by {a[ao"'" am' ·n. An r-place relation on an Jii' -definable subset of 1Jii'l n will be called Jii' -definable if its graph is an Jii'- nr definable subset of 1Jii' I . Now let L be a relational language such that xo' Xl ' ... are all the variables of L. We shall need the following theorem.

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Theorem 4.1. Given a k-dimensional interpretation a 1-+ a * of L in L", there corresponds to every L"-structure .PI an L-structure .PI * with I.PI * I = 1.PIlk such that (i) .PI * F a {:}.PI F a * for every closed a in L, (ii) every.PI *-definable set (relation) is .PI -definable.

We note that it makes sense to talk of .PI-definability of .PI *-definable sets because I.PI*I = 1.PIlk . We deal first with the case k = 1:

Lemma 4.1. Given a one-dimensional interpretation a 1-+ aO of L in L', there corresponds to every L'-structure .PI an L-structure .PI 0 such that I.PI°I = I.PII and (i) .PI 0 F a {:} .PI F aO for every closed a in L, (ii) every .PI°-definable set (relation) is .PI-definable.

Proof. We define the L-structure .PI 0 by requiring that for any n-place predicate symbol p of L, the corresponding n-place relation on .PI0, also denoted by p, satisfies (4.1) .PI °F p(a], ... , an) {::}.PI F (p(x]' ... , xn)) °[a], ... , an] for every a], ... , an E I.PI°I = I.PII. It is easy to see that (i) and (ii) are consequences of the following fact: for every formula a in L that has no free variables other than Xi ' ... , Xi ' I n

(4.2) .PI 0 F a [Xii Xin] {::}.PI F aD [Xii Xin] a] an a] an for arbitrary a], ... , an E I.PII. To show (4.2), note that when a is p(x] , ... , xn) , (4.2) coincides with (4.1). For other atomic formulas (i.e., a of the form p(x , ... ,X )), (4.2) is a consequence of (4.1) and (ii) in Definition I] In 2.1. Finally, to extend (4.2) to nonatomic formulas, we use (iii), Definition 2.1.

Lemma 4.2. If k 2: 1 and a 1-+ a(k) is the k-spread of L in L (k) , then to every L (k) -structure .PI there corresponds an L-structure .PI (k) such that I.PI (k) I = 1.PIlk and (i) .PI(k) F a {::}.PI F a(k) for every closed a in L, (ii) every .PI (k)-definable set (relation) is .PI-definable.

The proof is analogous to the proof of Lemma 4.1. For example, the L- structure .PI(k) is defined by the requirement that for any n-place predicate symbol p of L, the corresponding n-place relation on .PI(k) , also denoted by p , satisfies: .PI (k) FP ( a], ... ,an) {::}.PIFp (k) ( a]], ... ,a]k, ... ,an], ... ,ank ) for any ai = (ail' ... , aik ) E 1.PI(k) I = 1.PIlk, i = 1, ... , n.

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Combining Lemmas 4.1 and 4.2, we obtain Theorem 4.1 with s1'* = (s1'0)(k) . For a first application, let us denote by 9t the standard model of RCF, i.e., the field of real numbers viewed as an LOF-structure with universe 19t1 = R. Let (); f--+ (); * be the k-dimensional Cartesian interpretation of L peS in LOF (see end of §2). Then Theorem 4.1 taken with s1' = 9t , yields at place of s1'* the usual model ~ of E k . Definition 4.2. A structure s1' will be called semialgebraic if (SI) 1s1'1 = Rk for some k ~ 1, (S2) every s1' -definable set (relation) is 9t-definable.

Thus, by the preceding construction from Theorem 4.1, ~ is semialgebraic. Now consider an arbitrary k-dimensional interpretation (); f--+ (); * of L peS in L OF ' and let rp be a sentence in LpeS such that ReF f- rp* . Then 9t F rp* and Theorem 4.1 gives us a semialgebraic L PeS-structure 9t* such that 9t* F rp. It follows from this that to establish the main result of this paper, i.e., Theorem 1.1, we only need to prove the following:

Theorem 4.2. For every n ~ 2. there exists a theorem rpn of En such that the sentence rp n is false in every semialgebraic L peS-structure s1' with 1s1' I = Rk and 1 ::=:; k < n. The rest of this paper is devoted to establishing this result. We proceed as follows. In §5 we collect various known facts about 9t-definable (also called semialgebraic) sets and relations, and we introduce the concept of an n-tower in Rk (a system of subsets of Rk, Definition 5.1). We also prove that if 1 ::=:; k < n then there does not exist an n-tower in Rk (Theorem 5.1). In §6, we define the concept of an n-grid in a structure (Definition 6.1) and, if the structure s1' is semialgebraic, we associate with every n-grid in s1' an n-tower in 1s1' I (Theorem 6.1). Finally, in §7 we construct a theorem rp n of En such that for every semialgebraic L PeS-structure s1' , s1' F rp n implies the existence afan n-grid in s1' (Theorem 7.1). Obviously Theorems 5.1, 6.1, and 7.1 show that s1' F rpn cannot hold in a semialgebraic LpeS-structure s1' with 1s1' I = Rk and 1 ::=:; k < n.

5. SEMIALGEBRAIC SETS AND n-TOWERS For every n ~ 1, the class of semiaigebraic subsets of Rn is the smallest class of sets closed under forming complements, finite unions, and finite intersections, and containing all sets of the form {s E Rn : f(s) ~ O}, where s = (SI ' ... ,sn) and f is a polynomial in n variables with real coefficients. The well-known Seidenberg-Tarski theorem states that a set is semialgebraic if it is 9t -definable. To prove this, one views 9t as a model of the theory RCF and applies Tarski's theorem on the elimination of quantifiers from formulas of RCF. See [B, K-K, Se, TI], and a recent discussion in [VdD3]. Thus, without

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further reference to this result, we shall accept from now on gz -definable as equivalent to semialgebraic. A continuous map between semialgebraic subsets of Rm and RII is called semialgebraic if its graph is a semialgebraic subset of Rm x RII. According to Definition 4.1, an r-place relation on a semialgebraic subset of RII is semialgebraic if its graph is a semialgebraic subset of Rllr. The following properties of semialgebraic sets, maps, and relations will be needed later: I. The intersection of two semialgebraic sets is semialgebraic. II. The composite of semialgebraic maps (wherever defined) is semialgebraic. III. The direct and inverse image of a semialgebraic set under a semi algebraic map is semialgebraic. IV. Each semialgebraic equivalence relation on a semialgebraic set has a semialgebraic set of representatives. V. Each semialgebraic set has a finite semi algebraic triangulation. Properties I, II, and III follow without difficulty from Definition 4.1 of gz- definability. Property IV is shown in [VdDl, Proposition 4.3]. To state property V precisely, let us use il as a generic name for an open simplex in Rm :

where the vertices vo' ... , V d E Rm are not all contained in some hyperplane of dimension d - 1. We call d the dimension of il. Then a finite simplicial m complex in R is a set K = {ilj : j E J} , where J is finite and ilj are disjoint m open simplexes in R , such that all faces of each ilj also belong to K. In 1964, Lojasiewicz [L] proved the following: Triangulation theorem. Given a semialgebraic set S (in some RIl), there exists

a finite simplicial complex K = {ilj : j E J} (in some Rm) and, for some subset l' c J , a semialgebraic map

which is also a topological homeomorphism. Other proofs can be found in [C and H]. (The usually made assumption that S is bounded is not relevant for the previous version; see [H, Remark 1.10].) In a more general setting of O-minimal Tarski systems, this result is stated in [VdD2] and proved in [VdDl]. As a corollary we obtain Lemma 5.1. An infinite semialgebraic set contains an arc (homeomorphic image of a nonzero interval in R). We define now by recursion on n the n-towers under semialgebraic sets. Definition 5.1. (1) The I-towers under S are precisely the uncountable semialgebraic subsets of S .

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(2) If n ~ 2, then an n-tower T under S is a system (I:, II, T') , where (a) I: is any uncountable set, (b) II is a mapping that assigns to every b E I: a semi algebraic set II( b) c S so that II(b) nII(b') = 0 for every two distinct b, b' in I:, (c) T' is a mapping that assigns to every b E I: an (n - 1)-tower T' (b) under II( b) . An n-tower under a subset S of Rk will also be called simply an n-tower in Rk. It is easily seen that a set S under which there exists an n-tower is uncountable. Thus, for n ~ 2, each II( b) is uncountable and we can regard II( b) as a I-tower under itself. We conclude that if there is an n-tower in Rk with n ~ 2 , then there is also a 2-tower in Rk . Lemma 5.2. If n ~ 2, then there does not exist an n-tower in R. Proof. By the preceding remarks, it is enough to show that there does not exist a 2-tower in R. Now, if there were a 2-tower (I:, II, T') in R, we would have an uncountable family {II( b): b E I:} of uncountable, mutually disjoint, semialgebraiC subsets of R. By Lemma 5.1, each of these sets would have to contain an arc, thus also a rational number. However, we do not have uncount- ably many rational numbers.

Lemma 5.3. If S, Sl ,S2 are semialgebraic such that SCSI U S2 and there exists an n-tower under S, then there exists an n-tower under at least one of the intersections S n Sl' S n S2 . Proof. We use induction on n. If n = 1, we are given a I-tower under S, i.e., an uncountable semialgebraic subset S' of S. Then at least one of the sets S' n Sl ' S' n S2 is uncountable, and is thus a I-tower under itself. We assume now that n > 1 and that the assertion holds for all (n - 1)- towers. Let T = (I:, II, T') be an n-tower under SCSI U S2. Then for each b E I:, T' (b) is an (n - 1)-tower under II(b) C Sl U S2 ' so by the inductive assumption, there is an (n - 1)-tower under II( b) n Sl or under II( b) n S2 . Renaming the Si' if need be, we may assume that for each b E I:' , where I:' is some uncountable subset of I:, there is an (n - 1)-tower Til (b) under II(b)nSI . Denoting II(b)nSI by II'(b),weobtainan n-tower (I:' , II', Til) under S n S 1 • 0 Given a semialgebraic injective map h: S --> S' , one can associate in an obvious fashion with every n-tower T under S an n-tower h(T) under S' : the image of T by h. We shall use this observation several times in the subsequent proof. Lemma 5.4. If k, n ~ 2 and there exists an n-tower in Rk, then there exists an (n - I)-tower in Rk- 1 .

Proof. Let T = (I:, II, T') be an n-tower in Rk, where k, n ~ 2. As we observed earlier, each II(b) is uncountable. Thus, we have an uncountable

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family {II( b): bEl:} of nonempty, mutually disjoint, semialgebraic subsets of Rk. Obviously at least one of them has an empty interior (contains no open subset). Let this be II(b). By the triangulation theorem, we have a semialgebraic map and topological homeomorphism h: II(b) -+ U{L~j: j E i},

where I:l. j are open simplexes in some Rm • Consider the (n - 1)-tower T' (b) under II(b). Since II(b) = U{h- 1(1:l.): j E J'} and I:l.j and hence h-1(1:l.) are semialgebraic sets, we infer from Lemma 5.3 that there exists an (n - 1)- tower under one of the sets h -I (I:l.). Applying h, which is injective, we get I an (n - I)-tower under I:l.j • The restriction of h- to I:l.j is a topological homeomorphism onto h-I(I:l.) c II(b) c Rk, and since II(b) does not contain an open subset of Rk , the dimension of I:l.. is less than k, by the "invariance J . of domain theorem." See [E-S, p. 303]. Let d = diml:l.j ::; k - 1. We have I:l.j c Rm , so there is a linear map g: Rm -+ Rd such that the restriction gll:l.j is injective. Clearly g is semi algebraic and maps the just-obtained (n - 1)-tower under I:l.j onto an (n - I)-tower under g(l:l.) C Rd. Taking the image of this tower under the linear embedding Rd -+ Rk- I , we obtain an (n - 1)-tower in Rk - I • Theorem 5.1. If n > k , then there does not exist an n-tower in Rk . Proof. We use induction on k. The case k = 1 is Lemma 5.2. If k ~ 2, and there exists an n-tower in Rk for some n > k , then by Lemma 5.4, there also exists an (n - I)-tower in Rk- I • This provides the inductive step.

6. n-GRIDS IN STRUCTURES If we choose a rectangular coordinate system in ~, then the set of hyper- planes perpendicular to the coordinate axes is an example of an n-grid in ~ (see the example in this section). In general, we have: Definition 6.1. Let sf be a structure for a first-order language. A system (S, LI ' ... , Ln' PI ' ... , Pn), where n ~ 2, will be called an n-grid under S in sf if ( 1) S is an sf -definable subset of Isf I ; (2) L I , ... , Ln are any uncountable sets; (3) each Pi is a map from Li into the set of sf -definable subsets of S, such that Pi(b) n Pi(b') = 0 for every two distinct b, b' E Li and every i=I, ... ,n; (4) for arbitrary bi ELi (i = 1, ... , n),

PI (b l ) n P2(b2 ) n .. · n Pn(bn) =I- 0. Let us observe that in every n-grid all the sets Pi (b) are uncountable. Indeed, for any j =I- i, Pi(b) contains all the disjoint sets Pi(b) n Pj(a), where a E L j .

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Definition 6.2. Given an n-grid as above and a fixed a E L n , we shall call (Pn(a) , L I , ... ,Ln_ l , P;, ... , P~_I)' where P;(b) = PJb) n Pn(a) for each bE Li ; i ~ n - 1, the associated (n - I)-grid under Pn(a).

Example. For sf = ~ , let us define LI , ... ,Ln and Pi(b) as follows: Li is the ith coordinate axis Li = {(XI' ... , xn) E R':: Xj = °for all j =f. i}, and for each b E Li' Pi (b) is the (n - 1)-dimensional hyperplane passing through b and perpendicular to Li : Pi(b) = {(XI' ... , xn) ERn: (0, ... ,0, Xi' 0, ... ,0) = b}. Putting S = I~ I = Rn , we check easily that the conditions of Definition 6.1 are satisfied, except perhaps for ~ -definability; however, that will be obtained subsequently as a by-product of the proof of Theorem 7.1. Henceforth, we shall consider n-grids in semialgebraic Lpo-structures (see Definition 4.2). Theorem 6.1. Given a semialgebraic Lpo-structure sf , there is for each n 2:: 2 a mapping that assigns to every n-grid in sf an n-tower in Isf I, so that to a grid under S there corresponds a tower under S. Proof. We define the mappings by recursion on n. Let us consider an n-grid (S, LI ' ... , Ln , PI' ... ,Pn) in sf , where n 2:: 2. We assign to this n-grid an n-tower T = (l:, II, T') under S, where l: = L n , II( b) = Pn (b) for each bEl:, and T' is yet to be defined. Obviously l: is uncountable and II( b) is semialgebraic and also uncountable, as we observed just after Definition 6.1. Moreover: II(b) n II(b') = Pn(b) n Pn(b') = 0 for b =f. b' . Hence, if n = 2, we may take T' (b) = II( b) , which is a I-tower under itself, and we are done. If n > 2 then, by recursive assumption, there is already assigned to every (n - 1)-grid under S' in sf an (n - 1)-tower under S' . In particular, consider for each b E Ln = l:, the associated (see Definition 6.2) (n - I)-grid under Pn(b) , i.e., under II(b). By assumption, to this (n - I)-grid there is assigned an (n - 1)-tower under II(b). We denote this tower by T' (b) , thus completing the description of the n-tower T.

7. EXISTENCE OF n-GRIDS To complete the proof of Theorem 4.2 (and hence of Theorem 1.1; see end of §4) we still need to show the following: Theorem 7.1. For each n 2:: 2, there is a theorem 'Pn of En such that if sf is a semialgebraic Lpo-structure and sf F 'P n, then there exists an n-grid under S = Isf I in sf . Proof. Let sf be a semialgebraic Lpo-structure (fixed for the rest of this proof). Let us associate with any ao' al ' a2 E Isf I certain sf -definable sets, which we

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shall call by the names "line" and "hyperplane," since such names are normally used if ao' ai' a2 are distinct and .s:1' is ~. For this, we need two formulas of L pt5 which we shall denote by A and n. A is P(xO' Xl ' x2) V P(XI ' x2' xo) V P(X2' XO' Xl) and, if ao =I- a l ' the set {A[ao' a l ,.]} (see Definition 4.1, with parameter variables x O' Xl) gets the name "the line through ao' a l ." n is chosen so that for ao =I- ai' {n[ao' a l ' a2 , .]} should be called "the (n - 1)-dimensional hyperplane containing a2 and perpendicular to the line {A[ao' ai' .]}"; explicitly, we may take for n the formula:

3x43xs (Ie (X4 ' xs) & A [xo' Xl ' X4] & A [xo' Xl ' Xs] & J (X2 ' X4' X2' Xs) & J (X3 ' X4' X3' Xs)) .

Further, let P be the formula expressing "strict betweeness":

and let a be the conjunction of these five formulas: (1) 'v'xoe(xo'xo)' (2) 'v'xO'v'xl(e(xO,xl)-+e(xl'xO))' (3) 'v'XO'v'x l'v'x2(e(xO' Xl) & e(xl' X2 ) -+ e(xo' x2)), (4) 'v'xo'v'x I3x2(le(xo' Xl) -+ P[xo' X2 ' Xl))' (5) 'v'XO'v'XI 'v'X2'v'X3(P[xo ' X2 ' Xl] & P[xo' X3' X2] -+ P[xo' X3 ' Xl))'

Lemma 7.1. Suppose .s:1' ~ a and let ao' a l be such that .s:1' ~ le(ao' a l ) . l Then there exists an uncountable set L c {A[ao' ai' .]} such that .s:1' ~ le(b, b ) l l whenever b, b E.L and b =I- b .

Proof. It follows from .s:1' ~ a that e is an equivalence relation on 1.s:1' I. We show first that {A[ao' a l ' ·n contains infinitely many pairwise not e-equivalent elements. Let ao' ai' a2 , ••• be obtained from (4), so that .s:1' ~ p[ao' a i + l ' ail for i = 1,2, .... By the transitivity (5), we conclude that .s:1' ~ p[ao' a i + l ' a) for all j = 1,2, ... , i, which implies .s:1' ~ le(ai+ l ' aj ), by the definition of p. Also, putting j = 1, we get .s:1' ~ p[ao' ai+ l ,al ] whence all ai are on the required line. Obviously e is an .s:1' -definable relation, so it is semialgebraic. Restricting e to {A[ao' ai' .]} we get a semialgebraic equivalence relation on a semialgebraic set. Thus, property IV in §5 yields a semialgebraic set of representatives .L c {A[ao' a l ,·]} of the e-equivalence classes. As we have already shown, each of the ao' a l ' . .. is in a different equivalence class. Hence, .L is infinite. By Lemma 5.1, .L is uncountable. 0

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Evidently there exists a sentence r n in L po such that the meaning of st F r n is as follows:

THERE EXIST ao' al ' ••• , an such that: le(ai , a) for all distinct i, j = 0, ... , n AND for any 1 ::; i ::; n and every b, b' E {A, [ao' ai' .J} such that I e(b, b/) one has {n[ao' ai' b,·n n {n[ao' ai' b',·n = 0 AND for every bl ' ... , bn, if bi E {A. [ao' ai' .J} for 1 ::; i ::; n then {n [ao' ai' bl"J} n .. · n {n lao, an' bn"J} -10.

Let rp n be (J & r n . We have gn F r n (taking ao above to be the origin of a rectangular coordinate system in ~ and ai on the ith coordinate axis). Hence, En f- r n ' and since also En f- (J , we get En f- rp n ' as required. Suppose st F rp n ' and let ao' al ' •.• , an E 1st I have the properties assured by st F r n' For each i = 1, ... ,n we denote by Li an uncountable subset of {A,[ao' ai,·n such that st F le(b, b') for every two distinct b, b' ELi (see Lemma 7.1). Further, let S = 1st I and let Pi(b) = {n[ao' ai' b,·n for every b E Li and i = 1 , ... , n . Then, by st F r n ' the system (S, LI ' ... , Ln ' PI ' ... , Pn) satisfies all the conditions of Definition 6.1. Thus we have obtained an n-grid in st .

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[M-P-S] J. Mycielski, P. Pudhik, and A. S. Stern, A lattice 0/ chapters o/mathematics, Mem. Amer. Math. Soc. (to appear). [Se] A. Seidenberg, A new decision method/or elementary algebra, Ann. of Math. (2) 60 (1954), 365-374. [Sz] L. W. Szczerba, Dimensions o/interpretations (abstract), J. Symbolic Logic 49 (1984),677. [TI] A. Tarski, Decision method/or elementary algebra and geometry, 2nd ed., Berkeley, 1951. [T2] __ , What is elementary geometry?, Symposium on the Axiomatic Method with Special Reference to Geometry and Physics, North-Holland, Amsterdam, 1959, pp. 16-29.

DEPARTMENT OF MATHEMATICS, SULTAN QABOOS UNIVERSITY, SULTANATE OF OMAN

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